1. Introduction

Near-field radiative heat transfer has attracted increasing attention in the past decade since it offers heat flux far exceeding the blackbody limit and has found promising applications in addressing thermal management challenge in confined systems in a noncontact manner [1–8]. There have been a large body of works in the literature exploring various materials and geometries aiming to gain further heat transfer enhancement [3,9–14]. Three-body systems consisting of planar slabs have demonstrated considerably increased heat flux than their two-body counterparts, which was attributed to the excitation of cavity surface plasmon polaritons [15–19]. Recently, several studies reported three-body systems with periodic nanostructures engineered onto the bodies and the systems exhibited higher heat flux than the systems purely made of planar geometries [16,17]. These systems not only offered heat transfer amplification but also enabled thermal rectification due to the asymmetry [17,20–22]. Subsequent studies showed that the asymmetry could also be introduced by mismatched properties of materials, whose permittivity tensors were anisotropic. Such examples included applying bias voltage to graphene and exerting magnetic field to magneto-optical media [15,23,24]. Aside from the above active approach, rotation-induced modulation, which is passive, has also been investigated in two-body systems consisting of black phosphorus sheets [25]. As these systems removed the active control components, they may benefit from simplicity. In the past few years, Weyl semimetals gained increasing attention thanks to the intrinsic anomalous Hall effect [26,27] which enables nonreciprocal thermal emission and radiative heat transfer. Very recently, a thermal switch has been demonstrated in a two-body system consisting of bulk Weyl semimetal slabs [28]. The heat transfer could be substantially suppressed by the rotation-induced mismatch in the surface plasmon polariton modes and a maximum switch ratio of nearly 80% was shown. Considering the benefits offered by the three-body configuration, it would be interesting to investigate the three-body system made of Weyl semimetal slabs, in which up to three bodies can be simultaneously rotated and the heat transfer can be exceedingly enhanced due to the many-body effect. Furthermore, the temperature-dependent properties of Weyl semimetal may lead to phenomena that could not be observed in systems consisting of materials whose properties are independent of temperature. Additionally, it is also interesting to investigate scenarios where the position of the middle body is away from the center to induce asymmetric coupling between the two cavities.

In this work, we investigate the near-field heat transfer in a three-body system made of Weyl semimetals slabs. Compared with recent works of three-body systems based on symmetric or asymmetric gratings [16,17], our system has an advantage of geometry simplicity, which may benefit its practical implementation. The heat transfer is modulated by simultaneously rotating each body, in which the effect of the nonreciprocal surface plasmon polaritons is combined with the three-body effect. The heat transfer in the three-body system is significantly enhanced, however, the thermal switch ratio does not surpass that of the two-body counterpart. By further allowing asymmetric positioning of the middle body, the thermal switch ratio exceeds that of the Weyl semimetal two-body system.

2. Theoretical model

The Weyl semimetal three-body system we consider in this work is presented in Fig. 1. Three Weyl semimetal bodies are oriented along the z-axis, and we label the left, middle, and right bodies as bodies 1, 2, and 3, respectively. The regimes I to V are defined by their corresponding materials, i.e., the Weyl semimetals (I, III, and V) or vacuum (II and IV). Bodies 1 and 3 are semi-infinite plates at temperatures of $T + \mathrm{\Delta }{T_1}$ and $T - \mathrm{\Delta }{T_2}$, separated by vacuum gaps ${d_{12}}$ and ${d_{23}}$ from the body 2, which is at the temperature of T and has a thickness of ${t_2}$. The total distance between bodies 1 and 3 is then ${d_0} = {d_{12}} + {t_2} + {d_{23}}$. The near-field thermal radiation between two Weyl semimetal plates can be modulated by applying a relative rotation. Here, we assume the body 1 to be fixed at ${\theta _1} = \pi $ while bodies 2 and 3 can be freely rotated away from the x-axis in the xy-plane by angles of ${\theta _2}$ and ${\theta _3}$ ranging from 0 to $\pi $. As pointed out in Ref. [28], heat transfer coefficient in two-body Weyl semimetal system at different rotation angles, $h(\theta )$, is symmetric with respect to $\theta = \pi $. Based on this, in our calculation, we limit both rotation angles to be between 0 and $\pi $. We consider the incident electromagnetic waves parallel to the xz-plane with incident angles $\phi $ from the x-axis. We consider the case of three Weyl semimetal bodies that each hosts two Weyl nodes with a wavevector separation of $2\vec{{b}}$ in the momentum space along the y-axis, as indicated by k_y in Fig. 1. We further let $\vec{{b}} = b\vec{{y}}$ denote the orientation that no rotation has been implemented, such that $2\vec{{b}} ={-} 2b\vec{{y}}$ implies that the body 1 is rotated by $\pi $ (Fig. 1). With above definitions the permittivity tensor takes the form [29]

 

Fig. 1. Schematic of the investigated Weyl semimetal three-body system. We assume body 1 is the heat source and body 3 is the heat sink. Bodies 1, 2, and 3 are at temperatures of $T + \mathrm{\Delta }{T_1}$, T, and $T - \mathrm{\Delta }{T_2}$, respectively. Initially, the three bodies have the rotation angles $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,0,0} ]$. Body 1 is then fixed at $\pi $, while bodies 2 and 3 can be rotated around the z-axis from 0 to $\pi $. The system is split into 5 regions as labeled by Roman numbers, in which regions I, III, and V correspond to Weyl semimetals, whereas regions II and IV correspond to the vacuum. The total distance between bodies 1 and 3 is ${d_0} = {d_{12}} + {t_2} + {d_{23}} = 100$nm. Bodies 1 and 3 are semi-infinite in the z axis. $\textrm{E}({{k_x},{k_y}} )$ illustrates the electronic band structure of a Weyl semimetal with two Weyl nodes of opposite chirality separated by $2\vec{{b}}$ in the momentum space.

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In Eq. (2), ${\varepsilon _b}$ is the background permittivity, ${E_F}(T )$ is the chemical potential with the temperature dependence [31] captured by

For the three-body system illustrated in Fig. 1, the total heat flux ${Q_{tot}}$ from the body 1 to the body 3 can be evaluated as the summation of the heat fluxes from bodies 1 to 2, ${Q_{12}}$, and 1 to 3, ${Q_{13}}$, respectively. The expression is given by the fluctuational electrodynamics as [16,17]

In Eq. (4), the net heat flux flowing into the body 2 is defined by ${Q_{12}}$. Similarly, we can define the net heat flux flowing out of the body 2, ${Q_{23}}$, which quantifies the net energy exchange between the bodies 2 and 3. Note that at the thermal equilibrium, we have ${Q_{12}} = {Q_{23}}$. Following Eq. (5), we can write similarly the photon tunneling probability between the bodies 2 and 3 as, ${\xi ^{({2,3} )}} = \textrm{Tr}\{{{{\mathbb{D}}_{3,21}}{{{\mathbb W}}_1}({{{\mathbb{S}}_3}} ){\mathbb{D}}_{3,21}^\dagger [{{{{\mathbb W}}_{ - 1}}({{{\mathbb{S}}_{12 + }}} )- {{\mathbb{T}}_{2 + }}{{\mathbb{D}}_{1,2}}{{{\mathbb W}}_{ - 1}}({{{\mathbb{S}}_{\mathbf{1}}}} ){\mathbb{D}}_{1,2}^\dagger {\mathbb{T}}_{2 + }^\dagger } ]} \}$. With these quantities, the temperature of the body 2, $T = {T_2}$, can be uniquely determined by enforcing the thermal equilibrium condition, namely ${Q_{12}} = {Q_{23}}$, and solving for ${T_2}$ using an iterative process. To decouple the effect of the rotations of the bodies 2 and 3 from temperature-dependent properties of Weyl semimetal bodies, it is convenient to introduce the following heat transfer coefficients

We note that when assuming $\Delta {T_1} \to 0$ and $\Delta {T_2} \to 0$, the thermal equilibrium condition is naturally satisfied.

3. Results and discussions

In Fig. 2, we calculate the total heat transfer coefficient h at $T = 300\textrm{ K}$ as h₁₃ + h₂₃. Note that at thermal equilibrium, the heat fluxes in the two cavities are the same, i.e., Q₁₂ = Q₂₃ [16]. When $\Delta {T_1}$ and $\Delta {T_2}$ approach zero, which are what we assume here, it follows immediately that h₁₂ = h₂₃. Therefore, the total heat transfer coefficient can be equivalently expressed by h₁₃ + h₁₂ or h₁₃ + h₂₃. For convenience, we choose the latter. In Fig. 2(a), we set d₁₂ = d₂₃ = t₂= $\frac{{100}}{3}$ nm for simplicity. We assume that the optical properties of the body 2, which is much thinner than bodies 1 and 3, which are bulk, are the same as in its bulk form. By rotating bodies 2 and 3 each from 0 to $\pi $, we obtain pairs of ${\theta _2}$ and ${\theta _3}$, based on which we compute h. We identify two regions with apparently higher h values. The first region is localized near ${\theta _2} = \pi $, with the highest h located at $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$. First, we notice that the xz- and zx-components in the permittivity tensor ${\bar{\bar{\varepsilon }}_{xz}}$ and ${\bar{\bar{\varepsilon }}_{zx}}$[Eq. (1)] are maximized by the angles of 0 and $\pi $, leading to strong nonreciprocal evanescent waves. This is supported by the photon tunneling probability ${\xi ^{({1,3} )}}$ and ${\xi ^{({2,3} )}}$ calculated at the incident angle of 0 displayed in the first row of Fig. 2(c). Both ${\xi ^{({1,3} )}}$ and ${\xi ^{({2,3} )}}$ show high transmission of nonreciprocal evanescent modes outside the light line and these modes in ${\xi ^{({2,3} )}}$ extend to very large wave vectors $\beta $ (along the x-axis). ${\xi ^{({1,3} )}}$ shows high transmission of modes inside the light line, indicating that most propagating waves reach the body 3 through the body 2. In contrary, propagating modes in ${\xi ^{({2,3} )}}$ are limited which may be due to the small cavity size, i.e., d₂₃ = $\frac{{100}}{3}$ nm, over which evanescent waves dominate. Increasing the thickness of the body 2 or the cavity size, we can observe higher transmission for the propagating modes, as shown in Fig. 7 in the Appendix 5.2. At $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,\pi } ]$, h is relatively high but lower than $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$, even with maximized ${\bar{\bar{\varepsilon }}_{xz}}$ and ${\bar{\bar{\varepsilon }}_{zx}}$. This can be explained by the cancellation of nonreciprocal effect due to the mirror-symmetry in the system. It was found in [33] that reciprocal heat exchange function could be realized even between two nonreciprocal magneto-optical planar objects when the objects were subject to the same magnetic field. As the pair of Weyl nodes act similar to an internal magnetic field [32], we apply similar symmetry argument in our system in terms of the rotation angles implemented on the bodies since ${d_{12}} = {d_{23}}$. Specifically, for $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,\pi } ]$, ${\xi ^{({1,3} )}}$ is reciprocal and evanescent modes are significantly limited, whereas ${\xi ^{({2,3} )}}$ exhibits largely nonreciprocal evanescent modes towards large $\beta $ (see Fig. 8 and Fig. 9 in Appendix 5.3 for details about the symmetry). Thus, with reduced contribution from ${\xi ^{({1,3} )}}$, h gets smaller compared to that of $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$, yet the reduction is not substantial because ${\xi ^{({2,3} )}}$ still dominantly contributes. Similar analysis applies to the case of $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,0,\pi } ]$ (see Fig. 9(a) in Appendix 5.3), which locates on the diagonal defined by ${\theta _2} + {\theta _3} = \pi $. Moving down this diagonal, ${\bar{\bar{\varepsilon }}_{xz}}$ and ${\bar{\bar{\varepsilon }}_{xz}}$ decreases as ${\theta _2}$ and ${\theta _3}$ deviate from 0 or $\pi $. The mirror-symmetry in the system is broken and hence both ${\xi ^{({1,3} )}}$ and ${\xi ^{({2,3} )}}$ become nonreciprocal, as seen in the second row of Fig. 2(c) corresponding to $[{{\theta_1},{\theta_2},{\theta_3}} ]= \left[ {\pi ,\frac{\pi }{3},\frac{{2\pi }}{3}} \right]$. High ${\xi ^{({1,3} )}}$ is again limited to small $\beta $, while high ${\xi ^{({2,3} )}}$ persists at large $\beta $ although the frequency bandwidth narrows compared to the cases where ${\theta _2},{\theta _3} = 0$ or $\pi $. We now briefly discuss the “dark” regions having considerably small h, which are related to ${\theta _2},{\theta _3} = \frac{\pi }{2}$. These rotation angles lead to vanished ${\bar{\bar{\varepsilon }}_{xz}}$ and ${\bar{\bar{\varepsilon }}_{zx}}$. Note that ${\bar{\bar{\varepsilon }}_{xy}}$ and ${\bar{\bar{\varepsilon }}_{yx}}$ maximize, however, they are irrelevant to the nonreciprocal waves with respect to $\beta $ along the x-direction. The minimum h in Fig. 2(a) occurs at $[{{\theta_1},{\theta_2},{\theta_3}} ]= \left[ {\pi ,0,\frac{{5\pi }}{{12}}} \right]$ and we present ${\xi ^{({1,3} )}}$ and ${\xi ^{({2,3} )}}$ in the third row of Fig. 2(c). Both ${\xi ^{({1,3} )}}$ and ${\xi ^{({2,3} )}}$ are nonreciprocal, being consistent with the symmetry argument discussed above. In addition, the transmission of evanescent waves is significantly suppressed, which is due to small ${\bar{\bar{\varepsilon }}_{xz}}$ and ${\bar{\bar{\varepsilon }}_{zx}}$ with ${\theta _3}$ near $\frac{\pi }{2}$. This similarly applies to the case of $[{{\theta_1},{\theta_2},{\theta_3}} ]= \left[ {\pi ,\frac{\pi }{2},\pi } \right]$ despite that ${\xi ^{({1,3} )}}$ becomes reciprocal due to the mirror-symmetry in the system, as illustrated in the fourth row of Fig. 2(c). In the following, we keep d₀ = 100 nm unchanged and double the thickness of body 2, namely t₂ = $\frac{{200}}{3}$ nm, and hence d₁₂ = d₂₃ = $\frac{{50}}{3}$ nm. We calculate h similarly as done in Fig. 2(a) and the results are depicted in Fig. 2(b). We can see that, compared with Fig. 2(a), the rotation angle-dependence is almost the same, except that the magnitudes of h generally increase and is especially so along the diagonal. This can be attributed to increased contribution from nonreciprocal evanescent modes, as backed up by the photon tunneling probability in the first and second rows of Fig. 2(d), in which high transmission outside of the light line extends to much larger wave vectors compared with their counterparts in Fig. 2(c). The reduction of h shows strong dependence on the rotation angle near $\frac{\pi }{2}$ and seems less sensitive to the thickness of the body 2 [compare third and fourth rows in Figs. 2(c) and 2(d)].

 

Fig. 2. (a),(b) Heat transfer coefficients h calculated for rotation angles ${\theta _2}$ and ${\theta _3}$ each varied from 0 to $\pi $ while fixing ${\theta _1} = \pi $. (c), (d) Photon tunneling probabilities ${\xi ^{({1,3} )}}$ and ${\xi ^{({2,3} )}}$. In each subfigure, the first and second rows correspond to the systems with $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$ and $\left[ {\pi ,\frac{\pi }{3},\frac{{2\pi }}{3}} \right]$, while the third row corresponds to the system with $[{{\theta_1},{\theta_2},{\theta_3}} ]$ leading to the minimum h found in (a) and (b), i.e., $\left[ {\pi ,0,\frac{{5\pi }}{{12}}\; } \right]$, and the fourth row corresponds to the system with $[{{\theta_1},{\theta_2},{\theta_3}} ]= \left[ {\pi ,\frac{\pi }{2},\pi } \right]$. Body 2 has a thickness of ${t_2} = \frac{{100}}{3}$ nm for (a),(c), or ${t_2} = \frac{{200}}{3}$ nm for (b),(d).

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In Fig. 3, we show the effect of introducing asymmetry in the cavity size by shifting the position of body 2 with a thickness of $\frac{{100}}{3}$ nm, as schematically shown in the top panel of Fig. 3(g). We define ${{{d_{23}}} / {{d_{12}}}}$ to characterize the asymmetry. The cavity sizes between the bodies 1 and 2 and 2 and 3 are equal when ${{{d_{23}}} / {{d_{12}}}} = 1$; when the ratio is beyond 1, the right cavity (between the bodies 2 and 3) is larger than the left cavity (between the bodies 1 and 2), otherwise, it is smaller. We calculate h still at 300 K for ${{{d_{23}}} / {{d_{12}}}}$ ranging from $\frac{1}{3}$ to 19, as illustrated in Fig. 3(a)–3(e). The distribution of higher h values follows those observed in Fig. 2(a) and 2(b). The high h comes from the contribution of strong nonreciprocal evanescent waves at large wave vectors due to breaking the mirror-symmetry by both different rotation angles (already discussed in Fig. 2 for ${d_{12}} = {d_{23}}$) and the cavity size asymmetry. On one hand, when the body 2 is placed closer to the body 3, e.g., ${{{d_{23}}} / {{d_{12}}}} =$ $\frac{1}{3}$ as shown in Fig. 3(a), the left cavity gets larger and h is substantially enhanced compared with d₂₃ = d₁₂ [see Fig. 2(a)]. This is attributed to stronger coupling between bodies 2 and 3 as d₂₃ is reduced. On the other hand, when ${{{d_{23}}} / {{d_{12}}}} > 1$ the right cavity size increases, the regions [Fig. 3(b)–3(e)] corresponding to smaller h expands and becomes “darker”, suggesting h decreases towards greater cavity size asymmetry ${{{d_{23}}} / {{d_{12}}}}$. Note that we change the ranges of colorbars in Fig. 3(b)–3(e) to be different from Fig. 3(a) for better visualization. To elucidate what will happen when the asymmetry is extremely large, i.e., d₁₂ $\to 0$, we set d₁₂= 0, which changes the three-body system to a two-body system with the original body 2 acting as a coating being rotated by θ₂. For fair comparison, we adjust the geometrical parameters such that t₂ + d₂₃ = 100 nm. In such a system, we can see that the region with high h is greatly confined to ${\theta _2}$ near $\pi $. The maximum h appears at $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$ (or $[{{\theta_1},{\theta_3}} ]= [{\pi ,0} ]$ because bodies 1 and 2 have the same $\pi $ rotation and they are in contact), which is consistent with the symmetry argument and strong nonreciprocal effect resulting from the rotations that maximizes ${\bar{\bar{\varepsilon }}_{xz}}$ and ${\bar{\bar{\varepsilon }}_{zx}}$. And as indicated by the colorbar, h drops considerably, which is expected for the two-body system compared to its three-body counterparts. Interestingly, shifting the body 2 from the symmetric position along different directions allows not only the heat transfer suppression [Fig. 3(b)–3(e) vs. Fig. 2(a)] but also the heat transfer amplification [Fig. 3(a) vs. Fig. 2(a)]. In addition to these observations, we calculate the thermal switching ratio, defined by $\textrm{sw}({{\theta_2},{\theta_3}} )= {\left. {\frac{{{h_{\max }}({{\theta_2},{\theta_3}} )- {h_{\min }}({{\theta_2},{\theta_3}} )}}{{{h_{\max }}({{\theta_2},{\theta_3}} )}}} \right|_{{d_{23}}/{d_{12}}}}$, using the maximum and minimum h extracted from Fig. 3(a)–3(e), and plot the results in the bottom panel of Fig. 3(g). We emphasize that this definition clarifies that the switching ratio is calculated for fixed ${{{d_{23}}} / {{d_{12}}}}$ so that the change is induced solely by the rotations. As shown by the black markers, the switching ratio rises sharply at the beginning and gradually saturate beyond ${{{d_{23}}} / {{d_{12}}}}$= 10. One might expect such saturation to continue even for ${{{d_{23}}} / {{d_{12}}}}$$ \to \infty $, however, Fig. 3(f) and the resulting switching ratio instead suggest that the switching ratio will decrease. The saturation trend between ${{{d_{23}}} / {{d_{12}}}}$ = 10 and ${{{d_{23}}} / {{d_{12}}}}$= 20 [black markers in Fig. 3(g)] is due to the competing effects between the enhanced heat transfer enabled by the three-body system and the suppressed heat transfer arising from reducing the cavity size near the heat source (the body 1). When d₁₂ is extremely small, a three-body system would have the merit of heat transfer enhancement provided by the cavity surface plasmon polaritons and would approach a two-body system. The two effects mentioned above, which originally compete with each other, now both lead to heat transfer suppression. Interestingly, if we let the system start from the state in which ${d_{23}} = {{{d_{12}}} / 3}$ and $[{{\theta_2},{\theta_3}} ]= [{\pi ,0} ]$ [the rotation angles that yield the highest h in Fig. 3(a) and recall that ${\theta _1}$ is fixed at $\pi $] and changes to a final state yielding the smallest h for each ${{{d_{23}}} / {{d_{12}}}}$, or $\textrm{sw}({{\theta_2},{\theta_3}{{,{d_{23}}} / {{d_{12}}}}} )= \frac{{{{ {h({\pi ,0} )} |}_{{d_{23}} = {d_{12}}/3}} - {{ {{h_{\min }}({{\theta_2},{\theta_3}} )} |}_{{d_{23}}/{d_{12}}}}}}{{{{ {h({\pi ,0} )} |}_{{d_{23}} = {d_{12}}/3}}}}$, the maximum switching ratio found over this transition reaches 91% at ${{{d_{23}}} / {{d_{12}}}} = 19$, as illustrated by the green markers in Fig. 3(g). Note that we ignore the limiting case of ${{{d_{23}}} / {{d_{12}}}}$$ \to \infty $ here for fair comparison because apparently switching from a three-body system to two-body system would lead to larger switching ratio than switching between three-body systems with different configurations.

 

Fig. 3. Heat transfer coefficients h as a function of rotation angles ${\theta _2}$ and ${\theta _3}$ for cavity size asymmetries of (a) d₂₃ = $\frac{1}{3}$d₁₂, (b) d₂₃ = 3d₁₂, (c) d₂₃ = 7d₁₂, (d) d₂₃ = 15d₁₂, and (e) d₂₃ = 19d₁₂. In (f), we show the limiting case where d₁₂ is infinitesimally small, i.e., $\mathop {\lim }\limits_{{d_{12}} \to 0} \frac{{{d_{23}}}}{{{d_{12}}}} = \infty $, which can be considered as a two-body system with a coating of the thickness t₂ rotated by ${\theta _2}$. For fair comparison, we adjust geometrical parameters such that ${t_2} + {d_{23}} = {d_0} = 100$ nm. (g) Thermal switching ratio corresponding to (a) – (f), where the largest and smallest h values in each subfigure are extracted to be the numerator and denominator, respectively, in the case shown by the black markers. The green markers correspond to the ratio defined by switching from the initial state with $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$, and with d₂₃ = $\frac{1}{3}$d₁₂, to the final state having the minimum h due to rotations for d₂₃/d₁₂= $\frac{1}{3}$, 1, 3, 7, 11, 15, 19.

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So far, we have shown the effects of rotations and cavity size asymmetry on the thermal switching at 300 K with the assumption of infinitesimal temperature gradients. To reveal the effect of the temperature-dependent property of the Weyl semimetal, we impose a temperature gradient of 200 K by letting $T + \mathrm{\Delta }{T_1}$ = 400 K ($\mathrm{\Delta }{T_1} = $ 100 K) and $T - \mathrm{\Delta }{T_2}$ = 200 K ($\mathrm{\Delta }{T_2} = 100\; \textrm{K}$). As stated in [32], the Weyl semimetal has a small and nonconstant density of states due to its linear dispersion, which causes its chemical potential to be strongly temperature dependent, which can also be seen from Eq. (3). By changing the temperature from 200 K to 400 K, E_F calculated using Eq. (3) shows a change of ∼0.023 eV. This impacts ${\mathop{\rm Im}\nolimits} ({{\varepsilon_d}} )$ which is closely related to thermal radiation (see Fig. 10 in Appendix 5.4 for details). To show the effect of the temperature, in Fig. 4(b) and 4(c), we first assume $\mathrm{\Delta }{T_1} \approx 0$ and $\mathrm{\Delta }{T_2} \approx 0$, and compute h as a function of ${\theta _2}$ and ${\theta _3}$ (${\theta _1}$ is still fixed at $\pi $) when all three bodies are at 200 K and 400 K, respectively, as we did before. Despite the large variation in the magnitude of h, the rotation angle-dependence of h at the two temperatures show nearly identical characteristics. This implies that the effect of rotations on the heat transfer in our system dominates over that of the temperature for the temperature gradient of 200 K imposed here. Under thermal equilibrium condition, the equilibrium temperature of the body 2 can be uniquely determined via an iterative process to satisfy ${Q_{12}} = {Q_{23}}$. Ideally, to obtain the minimum heat flux at different $[{{\theta_2},{\theta_3}} ]$ for defining the thermal switch ratio, one needs to iteratively calculate equilibrium Q for $0 \le {\theta _2} \le \pi$ and $0 \le {\theta _3} \le \pi$. To allow manageable computational cost using our limited computing resources, based on the dominant influence of rotation angles observed in Fig. 4(b) and 4(c), we directly use those $[{{\theta_2},{\theta_3}} ]$ that lead to minimum h for varying cavity size asymmetry at 300 K identified in Fig. 3 (also tabulated in Table 1) and calculate the total heat flux Q contributed from two heat transfer channels, i.e., Q = Q₁₃ + Q₁₂ . When (Q₁₂ – Q₂₃)/Q₁₂ $\le $ 0.5%, we assume the thermal equilibrium is satisfied [16]. In Fig 4(d), we compute two sets of Q, namely for cases where the system has the maximum h given by $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$ (blue filled circles) and where the system has the minimum h given by $[{{\theta_1},{\theta_2},{\theta_3}} ]$ (red open circles) in Table 1. As the cavity size asymmetry ${{{d_{23}}} / {{d_{12}}}}$ increases, both sets exhibit peaks. For $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$, the peak is found at ${{{d_{23}}} / {{d_{12}}}}=1$, whereas for the other set the peak locates at ${{{d_{23}}} / {{d_{12}}}}=3$. The peak at ${{{d_{23}}} / {{d_{12}}}}=1$ for $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$ is expected because when the rotation angles of all three bodies are fixed, Q is dominated by the size of the two cavities which determine the coupling strengths for the two heat transfer channels Q₁₃ and Q₂₃. The couplings are maximized when the two cavities have the same size, which aligns with previous works [34,35]. The distribution of Q on the two sides of the peak is asymmetric because the heat transfer is nonreciprocal due to not only the lack of mirror-symmetry in the system as discussed in Fig. 2 but also the temperature-dependent permittivity of the bodies. For $[{{\theta_1},{\theta_2},{\theta_3}} ] = [{{\theta_1},{\theta_2},{\theta_3}} ]_{h=h_{min}}$, the peak slightly shifts to ${{{d_{23}}} / {{d_{12}}}}=3$. This is because $[{{\theta_1},{\theta_2},{\theta_3}} ] = [{{\theta_1},{\theta_2},{\theta_3}} ]_{h=h_{min}}$ is varied for each ${{{d_{23}}} / {{d_{12}}}}$ and thus Q is simultaneously influenced by the cavity size asymmetry and the rotations, which differs from the previous case (i.e., $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$). To support this observation, we further calculate Q by fixing $[{{\theta_1},{\theta_2},{\theta_3}} ]$ at $[\pi,0,5\pi/12 ]$, which are closed to $[\pi,0,\pi/2 ]$ that leads to small heat transfer coefficients when assuming infinitesimal temperature gradients as shown in Fig. 2 and Fig. 3, and explained in Fig. 9 in Appendix 5.3. We can see that the peak locates at ${{{d_{23}}} / {{d_{12}}}}=1$, which agrees with the peak location when $[{{\theta_1},{\theta_2},{\theta_3}} ]$ are fixed at $[{\pi ,\pi ,0} ]$. We can also observe that Q for $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$ is significantly larger than those of the other two sets, which is attributed to the strong excitation of nonreciprocal SPP modes in the system. We remark that Q at ${d_{23} \approx 0.07{d_{12}}}$ is only shown for $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$ to illustrate the decreasing trend at very small ${{{d_{23}}} / {{d_{12}}}}$. Such trends are expected for the other sets and we do not plot those Q in Fig. 4(d). Comparing with the two-body system, as shown by the dashed lines for respective $[{{\theta_1},{\theta_2},{\theta_3}} ]$, the heat flux Q dramatically increases in the three-body system, indicating the strong effect of inserting a middle body. The corresponding equilibrium temperatures T₂ of the body 2 corresponding to the three sets in Fig. 4(d) are illustrated in Fig. 4(e). The equilibrium temperatures for the three sets are very similar despite their substantially different Q. The trends follow the intuition that when the body 2 approaches the hot body (cold body) as ${{{d_{23}}} / {{d_{12}}}}$ increases (decreases), its equilibrium temperature T₂ goes up (goes down), which aligns with the existing literature [19,35–37]. Finally, in Fig. 4(f), we compute the thermal switching ratio. First, we use the thermal switch ratio definition $\textrm{sw}({{\theta_2},{\theta_3}} )= {\left. {\frac{{{Q_{\max }}({{\theta_2},{\theta_3}} )- {Q_{\min }}({{\theta_2},{\theta_3}} )}}{{{Q_{\max }}({{\theta_2},{\theta_3}} )}}} \right|_{{d_{23}}/{d_{12}}}}$ similar as in Fig. 3(g) to only focus on modulating the rotation angles for each cavity size asymmetry. Also note that Q_max occurs when $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$ is applied, thus the above definition is further simplified to $\textrm{sw}({{\theta_2},{\theta_3}} )= {\left. {\frac{{{Q}({{\pi},0} )- {Q_{\min }}({{\theta_2},{\theta_3}} )}}{{{Q}({{\pi},0} )}}} \right|_{{d_{23}}/{d_{12}}}}$. As shown by the empty-black-triangular markers (Fig. 4(f)), the switching ratio is overall below 0.6 and with the largest and smallest values found at ${{{d_{23}}} / {{d_{12}}}}=15$ and ${{{d_{23}}} / {{d_{12}}}}=7$. In addition, the switching ratios are much smaller than those provided by the optimal two-body system [gray dashed line in Fig. 4(f) under the same temperature conditions. Next, we use the switching ratio definition similar to that used in Fig. 3(g), namely $\textrm{sw}({{\theta_2},{\theta_3}{{,{d_{23}}} / {{d_{12}}}}} )= \frac{{{{ {Q({\pi ,0} )} |}_{{d_{23}} = {d_{12}}}} - {{ {{Q_{\min }}({{\theta_2},{\theta_3}} )} |}_{{d_{23}}/{d_{12}}}}}}{{{{ {Q({\pi ,0} )} |}_{{d_{23}} = {d_{12}}}}}}$ by fixing the initial state to $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$ and ${{{d_{23}}} / {{d_{12}}}}$ for each end state (i.e., each ${{{d_{23}}} / {{d_{12}}}}$) with $[{{\theta_1},{\theta_2},{\theta_3}} ] = [{{\theta_1},{\theta_2},{\theta_3}} ]_{h=h_{min}}$ as in Fig. 4(d). The initial state corresponds to the largest Q in Fig. 4(d). The redefined switching ratio is shown by green triangular markers [Fig. 4(f)]. This time, the switching ratio remarkably increases, with a maximum reaching 70% and the switching capability for ${{{d_{23}}} / {{d_{12}}}}$ >11 outperforms that of the optimal two-body system, which uses the rotation as the only tuning knob. This implies that by simultaneously tuning the rotation and the cavity size asymmetry, our system can achieve higher thermal switching performance. This comparison highlights the unique advantage of the three-body system over the two-body counterpart for thermal switching. Besides, since we have assumed that for the temperature gradient of 200 K the total heat flux minimizes at the rotation angles $[{{\theta_1},{\theta_2},{\theta_3}} ]_{h=h_{min}}$ found for h_min at 300 K (Fig. 3) due to the computational cost of scanning the entire space of $[{{\theta_1},{\theta_2},{\theta_3}} ]$ under the thermal equilibrium condition, the minimum heat flux may be found at other rotation angles and thus the switching ratio illustrated by the green triangular markers (Fig. 4(f)) may be slightly improved. As one example, we calculate the switching ratio using the set $[{{\theta_1},{\theta_2},{\theta_3}} ] = [\pi,0,5\pi/12 ]$ in Fig. 4(d) as the end state. As shown by the green diamond markers [Fig. 4(f)], the switching ratios are additionally enhanced due to the smaller Q at the end states. The maximum switching ratio reaches 75% (i.e., an increase of 5%) at ${{{d_{23}}} = 15{{d_{12}}}}$. This indicates the possibility of further improving the switching performance of our system for large temperature gradient by optimizing the rotation angles of the bodies at the thermal equilibrium.

 

Fig. 4. (a) Schematic of the case where a 200 K temperature gradient is imposed in the system. Heat transfer coefficient h as a function of rotations for (b) $T = 200$ K and (c) $T = 400$ K with $\mathrm{\Delta }{T_1} \approx 0$ and $\mathrm{\Delta }{T_2} \approx 0$ to illustrate the effect of temperature on h and its rotation angle-dependence. (d) Total heat flux at thermal equilibrium for the systems with varying cavity size asymmetry for different rotation angles. (e) Thermal equilibrium temperature of the middle body corresponding to each case in (d). (f) Thermal switching ratio calculated for three scenarios. Green triangles and diamonds represent scenarios where the denominator is fixed at the maximum h when d₂₃/d₁₂ = 1, whereas the empty-black triangles represent those in which the denominator is taken as the highest h for each d₂₃/d₁₂.

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Fig. 5. Photon tunneling probabilities calculated in the cavities between bodies 1 and 2 and between bodies 2 and 3 for $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$ with (a) ${{{d_{23}}} / {{d_{12}}}} = 1$ and (b) ${{{d_{23}}} / {{d_{12}}}} = 1/3$ and for (c) $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,0 ,0} ]$ with ${{{d_{23}}} / {{d_{12}}}} = 3$, and for (d) $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi, 0 ,5\pi}/12 ]$ with ${{{d_{23}}} / {{d_{12}}}} = 1/3$.

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Table 1. Combination of rotation angles of the bodies 2 (${{\mathbf \theta }_2}$) and 3 (${{\mathbf \theta }_3}$) used in Fig. 4(d)–4(f)

View Table

To gain more understanding about the trend of the total heat flux Q in Fig. 4(d) for the system with $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$ and $[{{\theta_1},{\theta_2},{\theta_3}} ] = [{{\theta_1},{\theta_2},{\theta_3}} ]_{h=h_{min}}$, we show the photon tunneling probabilities ${\xi ^{({1,2} )}}$ and ${\xi ^{({2,3} )}}$ in the two cavities at the incident angle of 0. As shown in Fig. 5(a), when $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$ and ${{{d_{23}}} / {{d_{12}}}}$=1, both ${\xi ^{({1,2} )}}$ and ${\xi ^{({2,3} )}}$ exhibit high transmission at large wave vectors, indicating the coupling strengths in both cavities are comparably strong, leading to the maximized heat flux Q as illustrated in Fig. 4(d). However, for the system with $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$ and ${{{d_{23}}} / {{d_{12}}}}=1/3$, as depicted in Fig. 5(b), ${\xi ^{({2,3} )}}$ shows high transmission for large wave vector modes, whereas those modes in ${\xi ^{({1,2} )}}$are considerably suppressed. This indicates that the coupling in the cavity between the bodies 2 and 3 is stronger, which agrees with its smaller size. When the system has $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,0 ,0} ]$ and ${{{d_{23}}} / {{d_{12}}}}=3$ (Fig. 5(c)), which leads to the peak Q for the $[{{\theta_1},{\theta_2},{\theta_3}} ] = [{{\theta_1},{\theta_2},{\theta_3}} ]_{h=h_{min}}$ set in Fig. 4(d), the modes at large wave vectors with high transmission appear in ${\xi ^{({1,2} )}}$, but they are significantly inhibited in ${\xi ^{({2,3} )}}$. This implies that the coupling in the cavity between the bodies 1 and 2 is dominant, which aligns well with its smaller size. Finally, in Fig. 5(d) we show one more case with the system set to $[{{\theta_1},{\theta_2},{\theta_3}} ] = [\pi,0,5\pi/12 ]$ and ${{{d_{23}}} / {{d_{12}}}}=1/3$. As expected, ${\xi ^{({2,3} )}}$ displays high transmission extending to large wave vectors, yet the transmission in ${\xi ^{({1,2} )}}$ is considerably lower, which is in agreement with the smaller cavity between the bodies 2 and 3 that provides greater coupling.

4. Conclusion

In summary, we have shown the near-field heat transfer in a three-body system made of Weyl semimetals. We found that the heat transfer is dominantly affected by the rotation of bodies despite the infinitesimal or large temperature gradient in the system. The heat transfer modulation strongly depends on the nonreciprocal surface plasmon polariton modes in cavities, based on which we have demonstrated thermal switching. The switching ratio is maximized by simultaneously tuning the rotations and the cavity size asymmetry and the maximum outperforms the optimal two-body system. This strategy applies to small and large temperature gradient cases and the former condition yields better switching performance. This work provides important understanding of the heat transfer in three-body systems and the resulting thermal switching based on tuning the nonreciprocal surface plasmon polaritons in Weyl semimetals via simultaneously controlling rotation angles and cavity sizes.

5. Appendix

5.1 Derivation of the reflection matrix

Here we derive the reflection matrices used in Eq. (5) and Eq. (6) in the main text. The Maxwell’s equation can be written in the compact form below,

(11)$${{\partial \psi (z )} / {\partial z = {{\mathbb K}}}}\psi (z ),$$

where $\psi (z )= {({{E_x},{E_y},{H_x},{H_y}} )^\textrm{T}}$ contains tangential electric and magnetic fields and ${{\mathbb K}}$ is a 4×4 matrix based on which the eigenvalues and eigenvectors can be determined. The solution to the differential equation in Eq. (11) can be written in the following form,

(12)$$\psi (z )= {{\mathbb W}}{e^{{\Lambda }z}}{({{{\mathbb{E}}_ - },{{\mathbb{E}}_ + }} )^\textrm{T}},$$

where ${{\mathbb W}} = {({{w_1},{w_2},{w_3},{w_4}} )}$ and ${\Lambda }$ are the eigenvector and eigenvalue matrices, respectively, and ${({{{\mathbb{E}}_ - },{{\mathbb{E}}_ + }} )^\textrm{T}}$ is a column vector representing the field amplitudes along the negative (-) and positive (+) z-directions.

We consider the system illustrated in Fig. 6, where a vacuum gap d₁₂ separates a semi-infinite and a finite-thick (thickness t₂) plate made of Weyl semimetals. Based on the material, we label four regions as “I”, “II”, “III”, and “IV”. The goal is to derive the ${{{\mathbb R}}_{12 + }}$ when I, II, and III are treated as an individual body. The fields in each region can be written by using Eq. (12) as ${\psi ^j}(z )= {{{\mathbb W}}^j}{e^{{{\overleftrightarrow {\Lambda }}^j}z}}{({{\mathbb{E}}_ -^j,{\mathbb{E}}_ +^j} )^\textrm{T}}$, where $j = $ I, II, III, and IV. At three interfaces formed at adjacent regions, the following boundary conditions are satisfied,

(13)$$\mathbf{III}/\mathbf{IV}:{{{\mathbb W}}^{\textrm{IV}}}\left( \begin{array}{l} {\mathbb{E}}_ -^{\textrm{IV}}\\ {\mathbb{E}}_ +^{\textrm{IV}} \end{array} \right) = {{{\mathbb W}}^{\textrm{III}}}\left( \begin{array}{l} {\mathbb{E}}_ -^{\textrm{III}}\\ {\mathbb{E}}_ +^{\textrm{III}} \end{array} \right),$$

(14)$$\mathbf{II}/\mathbf{III}:{{{\mathbb W}}^{\textrm{II}}}\left( \begin{array}{l} {\mathbb{E}}_ -^{\textrm{II}}\\ {\mathbb{E}}_ +^{\textrm{II}} \end{array} \right) = {{{\mathbb W}}^{\textrm{III}}}\left( {\begin{array}{{cc}} {{e^{ - i\overleftrightarrow {\Lambda }_ -^{\textrm{III}}{t_2}}}}&{}\\ {}&{{e^{ - i\overleftrightarrow {\Lambda }_ +^{\textrm{III}}{t_2}}}} \end{array}} \right)\left( \begin{array}{l} {\mathbb{E}}_ -^{\textrm{III}}\\ {\mathbb{E}}_ +^{\textrm{III}} \end{array} \right),$$

(15)$$\mathbf{I}/\mathbf{II}:{{{\mathbb W}}^{\textrm{II}}}\left( {\begin{array}{{cc}} {{e^{ - i{{\overleftrightarrow {k}}_{0z}}{d_{12}}}}}&{}\\ {}&{{e^{i{{\overleftrightarrow {k}}_{0z}}{d_{12}}}}} \end{array}} \right)\left( \begin{array}{l} {\mathbb{E}}_ -^{\textrm{II}}\\ {\mathbb{E}}_ +^{\textrm{II}} \end{array} \right) = {{{\mathbb W}}^\textrm{I}}\left( \begin{array}{l} {\mathbb{E}}_ -^\textrm{I}\\ {{\mathbb O}} \end{array} \right),$$

where ${\overleftrightarrow {\Lambda }_ - }$ and ${\overleftrightarrow {\Lambda }_ + }$ are 2×2 diagonal matrices containing eigenvalues in Weyl semimetal regions (i.e., I and III) corresponding to waves propagating along the – z- and + z-axis, respectively, and ${\overleftrightarrow {k}_{0z}}$ is a 2×2 diagonal matrix having eigenvalues in vacuum regions (i.e., II and IV) associating with waves moving along the + z-direction, and ${{\mathbb O}}$ is the 2×1 vector containing zeros. Rearranging Eq. (13)–(15) yielding,

(16)$$\left( \begin{array}{l} {\mathbb{E}}_ -^{\textrm{III}}\\ {\mathbb{E}}_ +^{\textrm{III}} \end{array} \right) = {[{{{{\mathbb W}}^{\textrm{III}}}} ]^{ - 1}}{{{\mathbb W}}^{\textrm{IV}}}\left( \begin{array}{l} {\mathbb{E}}_ -^{\textrm{IV}}\\ {\mathbb{E}}_ +^{\textrm{IV}} \end{array} \right),$$

(17)$$\left( \begin{array}{l} {\mathbb{E}}_ -^{\textrm{II}}\\ {\mathbb{E}}_ +^{\textrm{II}} \end{array} \right) = {[{{{{\mathbb W}}^{\textrm{II}}}} ]^{ - 1}}{{{\mathbb W}}^{\textrm{III}}}\left( {\begin{array}{{cc}} {{e^{ - i\overleftrightarrow {\Lambda }_ -^{\textrm{III}}{t_2}}}}&{}\\ {}&{{e^{ - i\overleftrightarrow {\Lambda }_ +^{\textrm{III}}{t_2}}}} \end{array}} \right){[{{{{\mathbb W}}^{\textrm{III}}}} ]^{ - 1}}{{{\mathbb W}}^{\textrm{IV}}}\left( \begin{array}{l} {\mathbb{E}}_ -^{\textrm{IV}}\\ {\mathbb{E}}_ +^{\textrm{IV}} \end{array} \right),$$

(18)$$\left( \begin{array}{l} {\mathbb{E}}_ -^{\textrm{II}}\\ {\mathbb{E}}_ +^{\textrm{II}} \end{array} \right) = \left( {\begin{array}{{cc}} {{e^{i{{\overleftrightarrow {k}}_{0z}}{d_{12}}}}}&{}\\ {}&{{e^{ - i{{\overleftrightarrow {k}}_{0z}}{d_{12}}}}} \end{array}} \right){[{{{{\mathbb W}}^{\textrm{II}}}} ]^{ - 1}}{{{\mathbb W}}^\textrm{I}}\left( \begin{array}{l} {\mathbb{E}}_ -^\textrm{I}\\ {{\mathbb O}} \end{array} \right),$$

 

Fig. 6. Schematic for deriving the reflection matrix considering bodies 1 and 2 as one unit. The Roman numbers label the respective media in each region, with I and III corresponding to Weyl semimetals and II and IV to vacuum.

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Substituting Eq. (18) into Eq. (17), we obtain the following relation between the fields in I and IV,

(19)$$\left( \begin{array}{@{}l@{}} {\mathbb{E}}_ -^{\textrm{IV}}\\ {\mathbb{E}}_ +^{\textrm{IV}} \end{array} \right) = {[{{{{\mathbb W}}^{\textrm{IV}}}} ]^{ - 1}}{{{\mathbb W}}^{\textrm{III}}}\left( {\begin{array}{{@{}cc@{}}} {{e^{i\overleftrightarrow {\Lambda }_ -^{\textrm{III}}{t_2}}}}&{}\\ {}&{{e^{i\overleftrightarrow {\Lambda }_ +^{\textrm{III}}{t_2}}}} \end{array}} \right){[{{{{\mathbb W}}^{\textrm{III}}}} ]^{ - 1}}{{{\mathbb W}}^{\textrm{II}}}\left( {\begin{array}{{@{}cc@{}}} {{e^{i{{\overleftrightarrow {k}}_{0z}}{d_{12}}}}}&{}\\ {}&{{e^{ - i{{\overleftrightarrow {k}}_{0z}}{d_{12}}}}} \end{array}} \right){[{{{{\mathbb W}}^{\textrm{II}}}} ]^{ - 1}}{{{\mathbb W}}^\textrm{I}}\left( \begin{array}{@{}l@{}} {\mathbb{E}}_ -^\textrm{I}\\ {{\mathbb O}} \end{array} \right),$$

The 4×4 eigenvector matrices in Eq. (19) may be rewritten as 2×2 block matrices as below,

(20)$${{{\mathbb W}}^\textrm{I}} = \left( {\begin{array}{{@{}cc@{}}} {{{\mathbb W}}_{\textrm{E}, - }^\textrm{I}}&{{{\mathbb W}}_{\textrm{E}, + }^\textrm{I}}\\ {{{\mathbb W}}_{\textrm{H}, - }^\textrm{I}}&{{{\mathbb W}}_{\textrm{H}, + }^\textrm{I}} \end{array}} \right),{{{\mathbb W}}^{\textrm{II}}} = \left( {\begin{array}{{@{}cc@{}}} {\overleftrightarrow {\mathbf k}_{0z}^{\textrm{II}}}&{\overleftrightarrow {\mathbf k}_{0z}^{\textrm{II}}}\\ {{{\mathbb{Q}}^{\textrm{II}}}}&{ - {{\mathbb{Q}}^{\textrm{II}}}} \end{array}} \right),{{{\mathbb W}}^{\textrm{III}}} = \left( {\begin{array}{{@{}cc@{}}} {{{\mathbb W}}_{\textrm{E}, - }^{\textrm{III}}}&{{{\mathbb W}}_{\textrm{E}, + }^{\textrm{III}}}\\ {{{\mathbb W}}_{\textrm{H}, - }^{\textrm{III}}}&{{{\mathbb W}}_{\textrm{H}, + }^{\textrm{III}}} \end{array}} \right),{{{\mathbb W}}^{\textrm{IV}}} = \left( {\begin{array}{{@{}cc@{}}} {\overleftrightarrow {\mathbf k}_{0z}^{\textrm{IV}}}&{\overleftrightarrow {\mathbf k}_{0z}^{\textrm{IV}}}\\ {{{\mathbb{Q}}^{\textrm{IV}}}}&{ - {{\mathbb{Q}}^{\textrm{IV}}}} \end{array}} \right),$$

where ${\mathbb{Q}} = \left( {\begin{array}{{cc}} 0&{ - i + {{i{\beta^2}} / {{k_0}}}}\\ {i{k_0}}&0 \end{array}} \right)$. For simplicity, we further define a matrix ${\mathbb{M}}$,

(21)$${\mathbb{M}} = {[{{{{\mathbb W}}^{\textrm{IV}}}} ]^{ - 1}}{{{\mathbb W}}^{\textrm{III}}}\left( {\begin{array}{{cc}} {{e^{i\overleftrightarrow {\Lambda }_ -^{\textrm{III}}{t_2}}}}&{}\\ {}&{{e^{i\overleftrightarrow {\Lambda }_ +^{\textrm{III}}{t_2}}}} \end{array}} \right){[{{{{\mathbb W}}^{\textrm{III}}}} ]^{ - 1}}{{{\mathbb W}}^{\textrm{II}}}\left( {\begin{array}{{cc}} {{e^{i{{\overleftrightarrow {k}}_{0z}}{d_{12}}}}}&{}\\ {}&{{e^{ - i{{\overleftrightarrow {k}}_{0z}}{d_{12}}}}} \end{array}} \right){[{{{{\mathbb W}}^{\textrm{II}}}} ]^{ - 1}},$$

such that Eq. (19) can be expressed as follows,

(22)$$\left( \begin{array}{l} {\mathbb{E}}_ -^{\textrm{IV}}\\ {\mathbb{E}}_ +^{\textrm{IV}} \end{array} \right) = {\mathbb{M}}{{{\mathbb W}}^\textrm{I}}\left( \begin{array}{l} {\mathbb{E}}_ -^\textrm{I}\\ {{\mathbb O}} \end{array} \right) = \left( {\begin{array}{{cc}} {{{\mathbb{M}}_{11}}}&{{{\mathbb{M}}_{12}}}\\ {{{\mathbb{M}}_{21}}}&{{{\mathbb{M}}_{22}}} \end{array}} \right)\left( {\begin{array}{{cc}} {{{\mathbb W}}_{\textrm{E}, - }^\textrm{I}}&{{{\mathbb W}}_{\textrm{E}, + }^\textrm{I}}\\ {{{\mathbb W}}_{\textrm{H}, - }^\textrm{I}}&{{{\mathbb W}}_{\textrm{H}, + }^\textrm{I}} \end{array}} \right)\left( \begin{array}{l} {\mathbb{E}}_ -^\textrm{I}\\ {{\mathbb O}} \end{array} \right),$$

After simple algebraic manipulations, we arrive at,

(23)$$\left( \begin{array}{l} {\mathbb{E}}_ -^{\textrm{IV}}\\ {\mathbb{E}}_ +^{\textrm{IV}} \end{array} \right) = {\mathbb{M}}{{{\mathbb W}}^\textrm{I}}\left( \begin{array}{l} {\mathbb{E}}_ -^\textrm{I}\\ {{\mathbb O}} \end{array} \right) = \left[ \begin{array}{l} ({{{\mathbb{M}}_{11}}{{\mathbb W}}_{\textrm{E}, - }^\textrm{I} + {{\mathbb{M}}_{12}}{{\mathbb W}}_{\textrm{H}, - }^\textrm{I}} ){\mathbb{E}}_ -^\textrm{I}\\ ({{{\mathbb{M}}_{21}}{{\mathbb W}}_{\textrm{E}, - }^\textrm{I} + {{\mathbb{M}}_{22}}{{\mathbb W}}_{\textrm{H}, - }^\textrm{I}} ){\mathbb{E}}_ -^\textrm{I} \end{array} \right],$$

The incident and reflected fields ${\mathbb{E}}_ - ^{\textrm{IV}}$ and ${\mathbb{E}}_ + ^{\textrm{IV}}$ can be linked as,

(24)$${\mathbb{E}}_ + ^{\textrm{IV}} = ({{{\mathbb{M}}_{21}}{{\mathbb W}}_{\textrm{E}, - }^\textrm{I} + {{\mathbb{M}}_{22}}{{\mathbb W}}_{\textrm{H}, - }^\textrm{I}} ){({{{\mathbb{M}}_{11}}{{\mathbb W}}_{\textrm{E}, - }^\textrm{I} + {{\mathbb{M}}_{12}}{{\mathbb W}}_{\textrm{H}, - }^\textrm{I}} )^{ - 1}}{\mathbb{E}}_ - ^{\textrm{IV}} = {{{\mathbb R}}_{12 + }}{\mathbb{E}}_ - ^{\textrm{IV}},$$

The reflection matrix ${{{\mathbb R}}_{12 + }}$ can then be obtained by,

(25)$${{{\mathbb R}}_{12 + }} = ({{{\mathbb{M}}_{21}}{{\mathbb W}}_{\textrm{E}, - }^\textrm{I} + {{\mathbb{M}}_{22}}{{\mathbb W}}_{\textrm{H}, - }^\textrm{I}} ){({{{\mathbb{M}}_{11}}{{\mathbb W}}_{\textrm{E}, - }^\textrm{I} + {{\mathbb{M}}_{12}}{{\mathbb W}}_{\textrm{H}, - }^\textrm{I}} )^{ - 1}}.$$

Other reflection (transmission) matrices in Eqs. (5),(6) in the main text can be derived through following similar processes described by Eqs. (11)–(25).

5.2 Photon tunneling probability for thicker middle body or larger cavity size

We provide additional examples of photon tunneling probability for thicker middle body and/or larger cavity size than the one chosen in the main text. Specifically, as also labeled in Fig. 7, we select two sets of parameters, namely d₁₂ = d₂₃ = 100 nm with t₂ = 100 ∼ 400 nm, and d₁₂ = d₂₃ = $\frac{{100}}{3}$ nm with t₂ = 100 ∼ 400 nm. Note that the second set is the system in Fig. 2 in the main text with larger cavity sizes. For simplicity, we show ${\xi ^{({1,3} )}}$ and ${\xi ^{({2,3} )}}$ at the incident angle of 0 and at the temperature of 300 K. For a larger cavity size of 100 nm, increasing t₂ decreases the transmission of propagating photons from the bodies 1 to 3 [${\xi ^{({1,3} )}}$], as seen in Fig. 7(a)–7(c), whereas the tunneling of propagating photons increases for ${\xi ^{({2,3} )}}$. Similarly, in Fig. 7(d)–7(f), with the smaller cavity size of $\frac{{100}}{3}$ nm, for ${\xi ^{({1,3} )}}$, larger t₂ leads to reduced transmission of propagating modes beyond the light line and also less excitation of evanescent modes at larger wave vectors. On the other hand, for ${\xi ^{({2,3} )}}$, higher transmission of evanescent modes is observed together with increased tunneling probability for propagating modes.

 

Fig. 7. Photon tunneling probabilities ${\xi ^{({1,3} )}}$ and ${\xi ^{({2,3} )}}$ computed at the incident angle of 0 for two sets of geometrical parameters. (a) – (c), the body 2 has a thickness ranging from 100 nm to 400 nm and the cavity sizes are d₁₂= d₂₃ = 100 nm. (d) – (f), the body 2 has a thickness ranging from 100 nm to 400 nm and the cavity sizes are d₁₂= d₂₃ = $\frac{{100}}{3}$ nm.

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5.3 Symmetry in the three-body Weyl semimetal system

Below, we describe the mirror-symmetry exists in the three-body Weyl semimetal system. The total heat transfer in the system is described by heat exchange functions (or photon tunneling probabilities) ${\xi ^{(2,3)}}({\omega ,\beta } )$ and ${\xi ^{(1,3)}}({\omega ,\beta } )$. When ${d_{12}} = {d_{23}} = {{{d_0}} / 3}$, the mirror-symmetry is satisfied for ${\xi ^{(1,3)}}({\omega ,\beta } )$ if bodies 1 and 3 have the same rotation angles, i.e., ${\theta _1} = {\theta _3}$, as depicted in the schematic in Fig. 8(e). In the presence of mirror-symmetry, ${\xi ^{(1,3)}}({\omega ,\beta } )$ is reciprocal, as seen in Fig. 8(g) and 8(h), otherwise, ${\xi ^{(1,3)}}({\omega ,\beta } )$ is nonreciprocal, as illustrated in Fig. 8(f). However, because ${\xi ^{(2,3)}}({\omega ,\beta } )$ is related to the heat transfer between bodies 2 and 3 and mirror-symmetry cannot be satisfied unless ${d_{12}} = 0$ and ${\theta _1} = {\theta _2} = {\theta _3}$, as shown in the schematic in Fig. 8(a). Further because we study the three-body system in which ${d_{12}} \ne 0$. Therefore, mirror-symmetry cannot be established for ${\xi ^{(2,3)}}({\omega ,\beta } )$ and consequently, as shown in Fig. 8(b)–8(d), ${\xi ^{(2,3)}}({\omega ,\beta } )$ is nonreciprocal.

 

Fig. 8. (a) Schematic illustration of the nonreciprocal exchange function ${\xi ^{(2,3)}}({\omega ,\beta } )$ due to the lack of symmetry. (b) – (d) Exchange functions ${\xi ^{(2,3)}}({\omega ,\beta } )$ calculated with different rotation angles and the incident angle $\phi $ of 0 for (a). (e) Schematic illustration of the reciprocal exchange function ${\xi ^{(1,3)}}({\omega ,\beta } )$ with the symmetry provided by ${\theta _1} = {\theta _3}$ and ${d_{12}} = {d_{23}}$. (f) – (h) Exchange functions ${\xi ^{(1,3)}}({\omega ,\beta } )$ calculated with different rotation angles and the incident angle $\phi $ of 0 for (e).

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Figure 9 serves to complement the explanation of the heat transfer coefficient h shown in Fig. 2(a) as a function of the rotation angles ${\theta _2}$ and ${\theta _3}$. When $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$, no mirror-symmetry exists in the system, and both ${\xi ^{(1,3)}}({\omega ,\beta } )$ and ${\xi ^{(2,3)}}({\omega ,\beta } )$ are nonreciprocal, leading to strongest contribution from nonreciprocal evanescent waves at wave vectors to the heat transfer. When $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,0,\pi } ]$ and $[{\pi ,\pi ,\pi } ]$, ${\xi ^{(1,3)}}({\omega ,\beta } )$ becomes reciprocal, and thus the heat transfer decreases. When $[{{\theta_1},{\theta_2},{\theta_3}} ]= \left[ {\pi ,\frac{\pi }{2},\pi } \right]$ and $\left[ {\pi ,0,\frac{\pi }{2}} \right]$, since one of the rotation angles is $\frac{\pi }{2}$, nonreciprocal evanescent wave of that body is significantly limited and cannot effectively couple to the nonreciprocal evanescent wave of other bodies, the heat transfer is substantially suppressed.

 

Fig. 9. (a)–(e) Exchange functions ${\xi ^{(2,3)}}$ and ${\xi ^{(1,3)}}$ at different rotation angles to qualitatively explain the variation of the heat transfer coefficient h colormap in the center [colorbar unit: W/m²/K].

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5.4 Temperature-dependent optical properties and heat transfer coefficients

Fig. 10(a) shows the chemical potential of the Weyl semimetal calculated using Eq. (3) from 0 K to 600 K, over which E_F drops from 0.163 eV to ∼0.11 eV. This change translates into the main diagonal permittivity component ${\varepsilon _d}$, whose real and imaginary parts are plotted in Fig. 10(b) and 10(c) for 200 K, 300 K, 400 K, and 600 K. Since the strength of the fluctuating dipoles that provide the source of thermal radiation is proportional to the imaginary part of the dielectric constant based on the fluctuation-dissipation theorem [38], ${\mathop{\rm Im}\nolimits} ({{\varepsilon_d}} )$ illustrated in Fig. 10(c) is more relevant to the near-field heat transfer. As temperature increases, we notice that ${\mathop{\rm Im}\nolimits} ({{\varepsilon_d}} )$ decreases. The influence on the heat transfer can be visualized by checking the exchange functions ${\xi ^{(1,3)}}$ and ${\xi ^{(2,3)}}$. As a concrete example, we choose the rotation angles $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$ and the incident angle of 0. As shown in Fig. 10(e) and 10(f), ${\xi ^{(1,3)}}$ and ${\xi ^{(2,3)}}$ are distinctly different at low and high temperatures. Specifically, for both of them, the nonreciprocal modes at high-frequency move closer to small wave vectors and the two bands gradually merge and covers broader frequency range. These seem to adversely affect the heat transfer coefficient h, however, when multiplied by the first derivative of the Bose-Einstein distribution $\Theta ^{\prime}(T )$, h increases with increasing T as expected, which is shown also in Fig. 10(d) for $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$ and an additional example for $[{{\theta_1},{\theta_2},{\theta_3}} ]= \left[ {\pi ,0,\frac{{5\pi }}{{12}}} \right]$.

 

Fig. 10. (a) Chemical potential E_F of the Weyl semimetal as a function of the temperature. (b) Real part and (c) imaginary part of the diagonal component of permittivity tensor of the Weyl semimetal ${\varepsilon _d}$ as a function of the temperature from 200 K to 600 K. (d) Heat transfer coefficient between 150 K to 450 K for two sets of rotation angles. Exchange functions (e) ${\xi ^{(1,3)}}$ and (f) ${\xi ^{(2,3)}}$ with $[{{\theta_1},{\theta_2},{\theta_3}} ]= [{\pi ,\pi ,0} ]$ and the incident angle of 0 for different temperature from 150 K to 450 K.

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Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable advice and critical comments.

Disclosures

The authors declare no conflicts of interest.

Data availability

The data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.

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